The carrot is just an item you get from Little Billy after his snowman is crushed by Manny from the tree. It's a Wimpy Wonderland object. It's not much of a useful item as you can put it on Fregley (stuck in the snowman).
Role In Wimpy Wonderland Island
After the Poptropican and Manny fall down the tree and smash Billy's snowman, this item is found sticking out of the ground. Pick it up. Later, when you have the leaf blower from the garage, go up to the snowman whose eyes follow you. Stick the carrot in in between it's eyes and it will say "Ouch! Hey, can you get me out of this thing?"
Trivia
- When you get Fregley out of the snowman, you can click "Use" on the item card, and he will still sy "Ouch! Hey, can you get me out of this thing?".